"Calculus" is a very broad term that I have seen with reference to all analytical maths. This includes complex analysis, real analysis, and even abstract algebra. Abstract algebra is a useful way of determining exactly how to perform operations like multiplication and division in very extreme circumstances, like as applied to quantum mechanics and our understanding of atomic (and subatomic) theory.
The analysis branches, from real analysis to complex analysis, help us engineer virtually everything. Risk assessment with financial investment is done with real analysis. Engineering approximations with series is done with real analysis. Materials science is done with complex analysis. Relativity is done with complex analysis. Etcetera, etcetera.
But this is all analysis. Let's use "calculus" in its narrower conception. It is the study of quantitative change.
It might seem obvious to you that numbers like \(\pi\) exist, or \(e\), both numbers incredibly important for things like engineering, science, finance, etc. But it's questionable whether they do actually exist. They're irrational numbers. They can't be expressed as a fraction \(\frac ab\mid\forall b\neq0\). For people at the time, grasping the concept of an irrational was much like grasping imaginaries today. How can we know they exist? What does it mean for something to exist in between fractions?
To definitively prove that \(\pi\) and \(e\) exist, we need to use calculus, because they are both directly related to rates of change. Like a ratio (fraction), but with less of a defined boundary. \(e\) for instance is a number whose rate of change is itself. That cannot be defined as a fraction. But it's used in virtually all electrical engineering (Fourier transforms are proved with concepts like \(e\)).
And I'd say electricity is pretty important.
Calculus many would argue is the first real math class a person takes. Everything before it is "common sense". I guess this makes math difficult to access, but it's a unfortunate property of the field. It gets much easier as you go along, though; like learning a language.

On these three you learn (probably first/second year of university)
1. Single variable calculus
2. Multivariable calculus/ Vector calculus
3. Differential equations (<--- this can be quite difficult)
I had
Partial Differential equations/Complex Variables Analysis/ A bit of differential geometry on my final year.